Inhomogeneous cubic congruences and rational points on del Pezzo surfaces (Q2842806)

Let NEWLINE\[NEWLINEX: x^2_0+ x_1 x^3_2+ x^3_1 x_3= 0NEWLINE\]NEWLINE be a surface in the weighted projective space \(\mathbb P(2,1,1,1)\), let NEWLINE\[NEWLINEU:= \{x\mid x\in X(\mathbb Z),\;\{x_0, x_1\}\subseteq \mathbb N,\;(x_1, x_2, x_3)= (1)\},NEWLINE\]NEWLINE and let NEWLINE\[NEWLINEN(B):= 2\text{\,card}\{x\mid x\in U,\, h(x)\leq B\},NEWLINE\]NEWLINE where NEWLINE\[NEWLINEh(x):= \max\{|x_0|^2, |x_1|, |x_2|, |x_3|\}\quad\text{for }x\in\mathbb R^4.NEWLINE\]NEWLINE The authors prove that NEWLINE\[NEWLINEN(B)= cB(\log B)^7(1+ O((\log\log B)^{-1/6})\quad\text{as }B\to\infty,NEWLINE\]NEWLINE with an explicitly given positive constant \(c\). Moreover, they remark that the minimal desingularisation \(\widetilde X\) of \(X\) is a del Pezzo surface of singularity type \(\mathbb E_7\), that \(\text{Pic\,}X\cong\mathbb Z^8\), and that, consequently, their asymptotic formula confirms the well-known conjectures, stated in the works of \textit{V. V. Batyrev} and \textit{Y. Tschinkel} [J. Algebr. Geom. 7, No. 1, 15--53 (1998; Zbl 0946.14009)], \textit{Yu. Manin} et al. [Invent. Math. 95, No. 2, 421--435 (1989; Zbl 0674.14012)], and \textit{E. Peyre} [Duke Math. J. 79, No. 1, 101--218 (1995; Zbl 0901.14025)]. Their proof involves counting suitably restricted integer points on the associated universal torsor, an open subset of an affine hypersurface in \(\mathbb A^{11}\). In the course of this counting, the authors are led to the rather complicated problem of obtaining a uniform upper bound for the number of solutions \((x,y)\) of the congruence \(ax^3+ by^3= 0(q)\) subject to the conditions NEWLINE\[NEWLINE1\leq x< X,\;|y|\leq Y,\;(xy,q)= (1),\;|ax^2+ by^3|\leq qB.NEWLINE\]